ENGINEERING SOCIETIES MONOGRAPHS THEORY OF Bakhmetef:Hydrauiics of Open Channeis Bleich:Buckling Strength of Metal Structures ELASTIC STABILITY Crandall:Engineering Analysis Elevatorski:Hydraulic Energy Dissipators Leontovich:Frames and Arches Nadai:Theory of Flow and Fracture of Solids Timoshenko and Gere:Theory of Elastic Stability Timoshenko and Goodier:Theory of Elasticity Timoshenko and Woinowsky-Krieger:Theory of Plates and Shells STEPHEN P.TIMOSHENKO Professor Emeritus of Engineering Mechanica American In urgical,and P tro Stanford Uniseraity snd the American Institute of Chemieal Engineers,have an arrangement with the McGraw-Hill Book Company,Ine.,for the production of a serics of selected books usefulness for engineers and industry. of thia are:to provide monographa of high technical IN COLLABORATION WITH manuricmisht not be publiahed sommercially because of too limited aale of engineeri ortant technical JAMES M.GERE without pecialirodtio to developmnrptolgaps nexi sting literature; to collect into one volume scattered informstion of especial timeliness on a given Associate Professor of Civil Engineering ocieties sasume no responsibility for any statementa made in these books. Stanford Unipersily xamined by one or more representa- tives of the societies competent to expressan opinion on the merita of the manuaeript. Ralph H.Phelps,CHAIRMAN Engineering Societies Library New York SECOND EDITION ENGINEERING SOCIETIES MONOGRAPHS COMMITTEE INTERNATIONAL STUDENT EDITION A.8.C.E. 且oward T.Critchlow H.Alden Foster EERING A.I.M.E. Nathanicl Arbiter John F.Elliott A.8.M.E. Calvin 8.Cronan Raymond D.Mindlin A.I.E.E. F.Maloolm Farmer Royal W.Sorensen MCGRAW-HILL INTERNATIONAL BOOK COMPANY A.I.Ch.E. Auckland Bogota Guatemala Hamburg Johannesburg Lisbon Joseph F.Skelly London Madrid Mexico New Delhi Panama Paris San Juan Charles E.Reed Sao Paulo Singapore Sydney Tokyo
PREFACE TO THE BECOND EDITION Numerous footnote references are given throughout the text as an aid to the student who wishes to pursue some aspect of the subject further. The authors take this opportunity to thank Mrs.Thor H.Sjostrand and Mrs.Richard E.Platt for assistance in preparing the manuscript and reading the proofs for this second edition. Stephen P.Timoshenko James M.Gere PREFACE TO THE FIRST EDITION The modern use of steel and high-strength alloys in engineering struc- tures,especinlly in bridges,ships,and aireraft,has made elastic insta- bility a problem of great importance.Urgent practical requirements have given rise in recent years to extensive investigations,both theo- retical and experimental,of the conditions governing the stability of such structural elements as bars,plates,and shells.It seems that the time has come when this work,recorded in various places and languages, often difficult of access to engineers who need it for guidance in design, should be brought together and put in the form of a book. The first problems of elastic instability,concerning lateral buckling of compressed members,were solved about 200 years ago by L.Euler.1 At that time the principal structural materials were wood and stone. The relatively low strength of these materials necessitated stout struc- tural members for which the question of elastic stability is not of primary importance.Thus Euler's theoretical solution,developed for slender bars,remained for a long time without practical application.Only with the beginning of extensive construction of steel railway bridges during the latter half of the past century did the question of buekling of com- pression members become of practical importance.The use of.steel led naturally to types of structures embodying slender compression mem- bers,thin plates,and thin shells. Experience showed that such struc- tures may fail in some cases not on account of high stresses,surpassing the strength of material,but owing to insufficient elastic stability of slender or thin-walled members. Under pressure of practical requirements,the problem of lateral buckling of columns,originated by Euler,has been extensively investi- gated theoretically and experimentally and the limits within which the theoretical formulas can be applied have been established.However, lateral buckling of compressed members is only a particular case of elaatie instability.In the modern design of bridges,ships,and aircraft we are confronted by a variety of stability problems.We encounter there not only solid struts,but built-up or "lattice-work"columns,and 1Loonard Euler's"Elnstie Curves,"tranalated and annotated by W.A.Oldfather, C.A.Ellis,and D.M.Brown,1933. vii
vi道 PREFACE TO THE FIRST EDITION PREPACE TO THE FIRST EDITION ix tubular members,where there is the possibility of local buckling,as well as buckling as a whole.In the use of thin sheet material,as in plate To the University of Michigan the author is grateful for financial girders and airplane structures,we have to keep in mind that thin plates support obtained from a research fund and used in the preparation of may prove unstable under the action of forces in their own planes,and numerical tables and diagrams for this book.He also takes this oppor- fail by buckling sideways.Thin cylindrical shells,such as vacuum tunity to extend thanks to Dr.D.H.Young who read over the complete vessels,which have to withstand uniform external pressure,may exhibit manuscript and made many valuable suggestions and corrections,to instability and collapse at a relatively low stress if the thickness of the Professors G.H.MacCullough and H.R.Lloyd who read some portions shell is too small in comparison with the diameter.The thin cylindrical of the manuscript,to Dr.I.A.Wojtaszak and Mr.S.H.Fillion for the shell may buckle also under axial compression,bending,torsion,or com- checking of equations and numerical tables,to Dr.Wojtaszak for reading binations of these.All such problems are of the utmost importance proofs,to Miss Reta Morden for the typing of the manuscript,and to in the design of airplanes of the modern monocoque type. Mr.L.S.Veenstra for the preparation of the drawings. S.Timoshenko In the discussion of these problems and their solutions,it has not been deemed necessary to include an account of the general theory of elastic stability,which finds its appropriate place in books on the theory of elasticity.This book proceeds directly to particular problems showing in each case under what conditions the question of stability calls for consideration.The various methoda of solution are presented in con- neetion with the types of problem to which they are best suited.The solutions have in most cases been supplemented by tables and diagrams which furnish values of critical loads and stresses for each particular case. While all available information relevant to a prescribed problem has been given,no attempt has been made to go beyond this into actual design,since it is a field in which other considerations besides rational theory and testing play their parts. The preliminary knowledge of mathematics and strength of materials taken for granted is that usually covered by our schools of engineering. Where additional mathematical equipment has been found necessary, it is given in the book with the appropriate explanations.To simplify the reading of the book,problems which,although of practical import- ance,are such that they can be omitted during a first reading are put in small type.The reader may return to the study of such problems after finishing the more essential portions of the book. Numerous references to papers and books treating stability problems are given in the book.These references may be of interest to engineers who wish to study some special problems in more detail.They give also a picture of the modern development of the theory of elastie sta- bility and may be of some use to graduate students who are planning to take their work in this field. In the preparation of this book the contents of a previous book dealing with stability problems and representing a course of lectures on the theory of thin plates and shells,as given in several Russian engi- neering schools,have been freely drawn upon. 1"Theory of Elasticity,"vol.II,St.Petersburg,Rusis,1916
CONTENTS Preface to the Second Edition. Preface to the First Edition Netations.. Chapter 1.Beam-columns..... 1 1.1 Introduction 12 Differentinl Equstions for Beam-oolumns 4 1 1.3 Beam-column with Concentrated Lateral Load 1 1.4 Several Coneentrated Loads 3 1.5 Continuous Lateral Losd. 4 79 1.6 Bending of s Beam-column by Couples 1.7 Approximate Formula for DeBections. 、 2 1.8 Beam-columns with Built-in Ends. 4 4 1.9 Beam-columns with Elastie Reatrainta 15 1.10 Continuous Beams with Axial Loads 1.11 Application of Trigonometrie Beries 1.12 The Effect of Initial Curvature on Deflections 4 ” 1.13 Determination of Allowable Stresses .. 3 Chapter 2.Elastic Buckling of Bars and Frames 6 2.1 Euler's Column Formuls 2.2 Alternate Form of the Differential Equntion for Determining Critical 6 Loads. 51 The Use of Beam-column Theory in Caleulating Critieal Loads 2.4 Buekling of Frames. 2.5 Buckling of Continuoun Beams 2.6 2.7 Buckling of Continuous Beams on Elastie Supporta 70 2.8 Large Defleetions of Buckled Bara(the Elnstics). The Energy Method. 2,9 Approximate Caleulntion of Critical Loads by the Energy Metbod 2.10 Buekling of a Bar on an Elastie Foundstion.. 2.11 Buckling of a Bar with Intermediate Compressive Forees. 86 2.12 Buckling of a Bar under Distributed Axinl Loads. 2.13 Buekling ofa Bar on an Elastic Foundation under Distributed Axial 100 2.14 Buekling of Bars with Changes in Croes Section 107 2.15 The Determination of Critical Loads by Buceesive Approximations. 113 2.16 Bars with Continuously Varying Crons Section. 116 2.17 The Effeet of Sbearing Force on the Critical Load. 132
CONTENTS 出 2.18 Buckling of Built-up Columns. 4 4 135 7.0 Buckling of Bimetallie Strip 310 2.19 Buckling of Helical Springs 4 7.10 Lateral Buckling of a Curved Bar with Cireular Axis 4 。4 313 2.20 Stability of n System of Bar8 2.21 The Case of Nonconservative Forees. Chapter 8.Bending of Thin Plates... 319 2.22 Stability of Priamatic Bars under Varying Axial Foreea 4 8.1 Pure Bending of Plates 4 319 82 Bending of Plates by Distributed Lateral Load. 163 326 Chapter 3.Inelastic Buckling of Bars 8.3 Combined Bending and Tension or Compression of Plates. 332 3.1 Inelastic Bending 163 8.4 Strain Energy in Bending of Plates 335 3.2 Inelastic Bending Combined with Axial Lond 8.5 Deflections of Rectangular Plntes with Simply Supported Edges. 340 3.3 Inelastic Buekling of Bars (Fundamental Case). g 8.6 Bending of Plates with a Small Initinl Curvature.. 844 3.4 Inelastie Buekling of Bars with Other End Conditions 182 8.7 Large Deflections of Plates. 346 Chapter 4.Experiments and Design Formulas 185 Chapter 9. Buckling of Thi扣ates, 348 4.I Column Tests 185 91 Methods of Caleulation of Critical Loads 348 4.2 Ideal-column Formulas as a Baais of Column Design 92 Buckling of Simply Bupported Rectangular Plates Uniformly Compreed 4.3 Empirical Formulas for Column Deaign. 路 in One Direction. 351 4.4 Ansumed Insecuracies as a Basis of Column Deaign 197 93 Buckling of Simply Supported Rectangular Plates Compreased in Two 4.5 Various End Conditions 202 Perpendieular Directions 4 a56 4.6 The Desigh of Built-up Columns. 206 9.4 Buckling of Uniformly Compreesed Reetangular Plates Simply Supported 212 along Two Opposite Sides Perpendicular to the Direetion of Compreasion Chapter 5.Torsional Buckling. and Having Various Edge Conditions along the Other Two Sides 4 360 5.1 Introduetion.. 212 9.5 Buekling of a Rectangular Plate Simply Supported along Two Opposite 52 Pure Torsion of Thin-walled Bars of Open Cross Section. 、 212 Sides and Uniformly Compressed in the Direction Parallel to Those Sides. 370 5.3 Nonuniform Torion of Thin-walled Bars of Open Croms Seetion.. 218 9.6 Buckling of a Simply Supported Rectangular Pinte under Combined 5.4 Torsional Buekling. 225 Bending and Comprension. 373 5.5 Buckling by Torsion and Flexure. 229 9,7 Buckling of Rectangular Plates under the Action of Shearing Btreascs 379 5.6 Combined Torsional and Flexural Buckling of a Bar with Continuous 9.8 Other Cnses of Buckling of Rectangular Plates 385 Elastic8 upporte,· 237 9.9 Buckling of Circular Platee. 380 5.7 Torsional Buckling under Thrust and End Moments 244 9.10 Buckling of Plates of Other Shapes 392 9.11 Stability of Plates Reinforeed by Ribs 39H Chapter 6. Lateral Buckling of Beams 4 251 9.12 Buckling of Plstes beyond Proportional Limit 408 6.1 Differential Equations for Lateral Buckling 251 9.13 Large Deflections of Buckled Plates 41 6.2 Lateral Buckling of Beams in Pure Bending 4 253 9.14 Ultimate Strength of Buckled Plates 418 6.3 Lateral Buckling of a Cantilever Beam 257 9.15 Experiments on Buckling of Platee 42 6.4 Lateral-Buckling of Simply Supported I Beams 262 9.16 Practical Applications of the Theory of Buckling of Piatee 29 6.5 Lateral Bucklins of Simply Supported Beam of Narrow Rectangular Chapter 10. Bending of Thin Shells.. 440 Cross Section. 268 6.8 Other Cases of Lateral Buekling. 270 10.1 Deformation of an Element of a Shell 440 6.7 Inelastie Lateral Buckling of I Beams 272 10.2 Symmetrical Deformation of a Cireular Cylindricnl Shell 10.3 Inextensional Deformation of Cireular Cylindrical Shell Chapter 7. Buckling of Rings,Curved Bars,and Arches. 278 445 10.4 General Case of Deformation of a Cylindrical Shell. 4M8 7.1 Bending of a Thin Curved Bar with a Circular Axis 278 10.5 Symmetrical Deformation of a Bpberical Shell 4 453 72 Application of Trigonometrie Series in the Anslysis of a Thin Circular Chapter 11.Buckling of Shells 457 ing· 44 ” 282 73 Effect of Uniform Preasure on Bending of a Cireulnr Ring.. 287 111 Symmetrical Buekling of a Cylindrienl Shell under the Action of Uniform 7.4 Buckling of Cireular Rings and Tubes under Uniform External Proasure 289 Axial Compression 457 7.5 The Design of Tubee under Uniform External Pressure on the Basis of 11.2 Inextensional Forms of Bending of Cylindrical Shells Due to Instability 461 A题mmed Inaecuracies + 294 11.3 Buekling of a Cylindrical Shell under the Action of Uniform Axial Pressure 462 7.6 Buckling of a Uniformly Compressed Cireular Arch ▣44 11.4 Experimenta with Cylindrical Shells in Axial Compreasion 488 7,7 Arebes of Other Forms. 11.5 Buckling of a Cylindrical Shell under the Action of Uniform Externsl 7.8 Buekling of Very Flnt Curved Bars .... 。4 305 Lateral Preeeure. 474